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Questions and Answers
What is the domain of the composition (f o n)(x)?
What is the domain of the composition (f o n)(x)?
Which function produces the expression 1/x²?
Which function produces the expression 1/x²?
What is the final expression for (f o m)(x)?
What is the final expression for (f o m)(x)?
Which composition produces the quadratic expression (x² - 10x + 25)?
Which composition produces the quadratic expression (x² - 10x + 25)?
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Which function composition equals x?
Which function composition equals x?
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Study Notes
Composite Function Domains
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Steps to find the domain of a composite function:
- Find the domain of the inner function.
- Find the domain of the composite function (result).
- Ensure the final domain includes restrictions from both steps (a) and (b).
Function Compositions and Domains
- f(x) = √x, g(x) = √x, h(x) = 1/x
- p(x) = x² - 10x + 25
- n(x) = x + 5
Examples of Function Composition
- (f o n)(x) = f(n(x)) = √(x + 5). Domain: x ≥ -5
- (h o p)(x) = h(p(x)) = 1/(x² - 10x + 25). Domain: x ≠ 5
- (f o m)(x) = f(m(x)) = f(x²) = √x² = |x|. Domain: all real numbers
- (h o h)(x) = h(h(x)) = h(1/x) = 1/(1/x) = x. Domain: x ≠ 0
Identifying Composite Functions
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Expression: x² - 10x + 25
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Composition: p(m(x))
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Composition: p(m(x))
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Expression: √x² + 5
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Composition: g(m(x))
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Description
This quiz covers the concepts of composite functions and how to determine their domains. Explore various examples involving square roots, polynomials, and rational functions to solidify your understanding of the composition process. Test your knowledge on finding domains while considering restrictions from inner and outer functions.
